Thanks to the Schläfli identity, \(\sum _{e}^6 j_l \frac{\partial \Theta _l}{\partial j_l}=0\) , the phase term has a simple derivative expression \(\frac{\partial \sum _{l}j_{l}\Theta _l}{\partial j_l}= \Theta _l\) . The saddle point \(\frac{\partial {\mathcal {S}}_\epsilon }{\partial j_l }\)  can be separated into the real part and the imaginary part.
Using some trigonometry relations, the result reads (neglecting sub-leading contributions)[1]}, [2]}
\(\frac{\partial \text{Re}[{\mathcal {S}}_\epsilon ]}{\partial j_e}=0 \,\,&\rightarrow \,\,|Y_l|^2\simeq \sqrt{\frac{k_{ll_1}k_{ll_2}}{k_{l_1l_2}J_{s(l)}}\frac{k_{l\tilde{l}_1}k_{l\tilde{l}_2}}{k_{\tilde{l}_1\tilde{l}_2}J_{t(l)}}}\equiv \tan \frac{\phi _{s(l)}}{2} \tan \frac{\phi _{t(l)}}{2}\,, \\\frac{\partial \text{Im}[{\mathcal {S}}_\epsilon ]}{\partial j_l}=0 \,\,&\rightarrow \,\,\Psi _l = -\frac{\epsilon }{2}\Theta _l\,. \) 
